RI: Small:Differential Ray Geometry for Surface Reconstruction and Modeling

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This project focuses on developing the theoretical foundation of differential ray geometry. The PI first derives a comprehensive ray geometry framework including the ray-distortion, ray-caustics, and ray-curvature theories as well as ray differential operators. This new framework is widely applicable to real-world problems. On the computer vision front, the PI explores robust and efficient schemes to infer ray structures from distortions or from caustics patterns and then recover surface geometry from ray differential attributes. This leads to a new class of specular (reflective and refractive) surface reconstruction algorithms. On the computer graphics front, the PI employs a novel normal-ray representation that converts a smooth 3D surface into a 2D ray manifold so that surface differential attributes can be directly derived from normal-ray geometry. The PI further develops new subdivision, re-meshing, and mesh simplification schemes for generating surfaces consistent with the underlying normal ray structures. This research benefits many computer vision and graphics applications by providing a differential ray geometry model for cameras, light sources, and surfaces. It also benefits shape designs in aircraft, automobile, and many other industries, where higher-order shape consistencies are required. This project contributes to education through the development of new differential geometry courses and seminars and by involving women and under-represented students in mathematical and computer science research. The PI further seeks to build strong connections with the fields of mathematics and physics, optical engineering, and mechanical engineering through the project.